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Riemann-Sieltjes vs. Lebesgue

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jsr9119
#1
Dec4-12, 08:56 PM
P: 14
Hey guys,

I'm doing a paper on the Radon transform and several sources I've come across cite the Lebesgue integral as a necessary tool to handle measures in higher order transforms.
But, Radon's original paper employs the Riemann-Stieltjes integral in its place.

I read that Lebesgue is more general and so Radon could have used it in place of RSI. Is this the case?

Thanks,

Jeff
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micromass
#2
Dec5-12, 08:07 AM
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The Lebesgue integral is indeed more general than the Riemann integral.
Using measure theory, we can also develop the Lebesgue-Stieltjes integral, and this is a generalization of the Riemann-Stieltjes integral.

So yes, the paper could probably be written with Lebesgue instead of Riemann. But there may be technical differences between the two.
Stephen Tashi
#3
Dec5-12, 10:15 AM
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Is there a distinction between "Lebesgue Integration" and "integration with respect to Lebesgue measure"? My impressions is that "Lebesgue measure" on the real number line is a particular measure that implements the usual notion of length, so the measure of a single point would be zero. On the other hand, is "Lebesgue integration" defined with respect to an arbitrary measure?

For Lebesgue Integration to include Riemann-Stieljes integration as a special case, is it necessary to use measures other than Legesgue measure? (I'm thinking of the specific example of defining an integration that can integrate a discrete probability density function by the method of assigning non-zero measure to certain isolated points and turning "integration" into summation.)

mathman
#4
Dec5-12, 03:10 PM
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Riemann-Sieltjes vs. Lebesgue

Lebesgue-Stieljes integral is best described as Lebesgue integration with respect to a given measure.
pwsnafu
#5
Dec5-12, 06:14 PM
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P: 820
Quote Quote by Stephen Tashi View Post
Is there a distinction between "Lebesgue Integration" and "integration with respect to Lebesgue measure"? My impressions is that "Lebesgue measure" on the real number line is a particular measure that implements the usual notion of length, so the measure of a single point would be zero. On the other hand, is "Lebesgue integration" defined with respect to an arbitrary measure?
Yes. Lebesgue integration is defined with respect to a measure. The procedure is the same, but different measures give different integrals.

For Lebesgue Integration to include Riemann-Stieljes integration as a special case, is it necessary to use measures other than Legesgue measure?
The Lebesgue measure is derived from the set function ##m((a,b])=b-a##.
The Stieljes measure is derived from the set function ##m(a,b]) = g(b)-g(a)## for some monotonically increasing function g.

(I'm thinking of the specific example of defining an integration that can integrate a discrete probability density function by the method of assigning non-zero measure to certain isolated points and turning "integration" into summation.)
Summation is a special case of Lebesgue integration, using the counting measure over Z, or Dirac measure over R.
micromass
#6
Dec5-12, 06:39 PM
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In my experience, it's a bit ambiguous. When talking about Lebesgue integration, sometimes people talk about general integration wrt a measure and sometimes they talk about integration wrt Lebesgue measure. It's usually clear from the context though.


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